Tiling asks whether polyominoes can cover a board exactly. Below: the area rule, the chessboard argument, tiny worked pictures, and free games that are tiling puzzles in disguise.
A polyomino tiling is an exact cover: every free cell belongs to exactly one piece. Packing puzzles on PuzzlePave are finite tilings with a fixed tray of pieces; solvers search the same space with dancing links.
Area first, then colouring
Dominoes need an even number of free cells.
Trominoes need a multiple of 3; tetrominoes of 4; pentominoes of 5.
Even with the right area, a colouring can prove impossibility — the classic example removes two same-colour corners from a chessboard.
4×4 domino tiling
Sixteen cells, eight dominoes — always possible on a plain rectangle.
Damaged board (sketch)
Opposite corners blocked: area stays even, but a chessboard colouring still forbids a complete domino tiling of a true mutilated board.
Small tilings you can see
Trominoes on 3×4
Twelve cells → four trominoes. Straight and V pieces share the work.
Squares as tetrominoes
Four 2×2 blocks tile a 4×4 — the same idea as our square-packing game.
Twelve pentominoes cover exactly 60 squares, so they tile 6×10, 5×12, 4×15 and 3×20 rectangles (thousands of ways). See the labelled 6×10 on the pentomino shapes page or browse solutions in the solver guides.
A four-step approach
Check the area. If free cells are not a multiple of the piece size, stop.
Colour the board. Look for imbalances each piece type cannot fix.
Seat awkward pieces. Narrow corridors and few-orientation shapes first.
Protect connectivity. Pockets must stay multiples of the piece size.
You cover every square of a board with polyomino pieces so that pieces do not overlap and do not hang off the board. That is an exact-cover problem — the same idea behind packing puzzles.
When can dominoes tile a board?
A board can be tiled by dominoes only if it has an even number of squares. The classic mutilated chessboard removes two opposite corners (same colour), leaving unequal black and white counts, so no domino tiling exists even though 62 is even.
What about trominoes and pentominoes?
Trominoes need a multiple of 3 squares; pentominoes a multiple of 5. Having the right area is necessary but not always enough — colouring arguments and narrow corridors can still block a tiling.
How is tiling related to PuzzlePave’s packing games?
Our packing levels are finite tiling instances: the piece set and board were generated so at least one tiling exists (checked with dancing-links). Hints reveal one piece of a known solution.
Where can I try polyomino tiling online?
Play the free packing puzzle, tromino, tetromino, hexomino or square-packing games, browse pentomino solutions, or open the packing / pentomino solvers.